Problem: A huge ice glacier in the Himalayas initially covered an area of $45$ square kilometers. Because of changing weather patterns, this glacier begins to melt, and the area it covers begins to decrease exponentially. The relationship between $A$, the area of the glacier in square kilometers, and $t$, the number of years the glacier has been melting, is modeled by the following equation. $A=45e^{-0.05t}$ How many years will it take for the area of the glacier to decrease to $15$ square kilometers? Give an exact answer expressed as a natural logarithm. years
Solution: Thinking about the problem We want to know how many years, $t$, it will take for the area of the glacier, $A$, to decrease to $15$ square kilometers. So we need to find the value of $t$ for which $A=15$. Substituting $15$ in for $A$ in the model gives us the following equation. $15=45e^{-0.05t}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}45e^{-0.05t}&=15\\\\ e^{-0.05t}&=\dfrac{1}{3}\\\\ -0.05t&=\ln\left(\dfrac{1}{3}\right)\\\\ t&=-20\cdot {\ln\left(\dfrac13\right)}\\\\ \end{aligned}$ It will take $-20\cdot {\ln\left(\dfrac13\right)}$ years for the area of the glacier to decrease to $15$ square kilometers. The expression above represents an exact solution to the equation. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $-20\cdot {\ln\left(\dfrac13\right)}$ years.